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3137 Data Structures and Algorithms in C++
Lecture 7July 26 2006
Shlomo Hershkop
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Announcements will do review later today
will take questions at end
please make sure to submit/plan hwsemester is going to end in 2 weeks from today
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OutlineSorting – quick sortDisjoint DSReview for midterm
Reading: Chapter 7.7-7.8, 8-8.3
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Quick sortfastest currently known sort
Average N log NWorst: N2
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Quicksortif one element returnelse
pick a pivot from the listsplit the list around the pivotreturn quicksort(left) + pivot + quicksort(right)
Lets do an example
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issuesHow does worst case happen ?
how to pick the pivot ??
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Pivot #1use the first element of the list
pro/cons ?
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sorted list will always be N2
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Pivot #2choose random element for pivot
pro/cons ?
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great performance
expensive to generate random number
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Pivot #3Choose median value from the list
pro/cons ?
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hmmm don’t you need a sorted list to get median?
actually there is a linear algorithm for this ☺ will be doing it on homework
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Pivot #4Median of 3
since #3 isnt cheap, can grab 3 elements and take median
can even use random if you don’t mind
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codingok so enough theory, how do you code all this ??
arrays are much cheaper than linked lists
lots of tricks to keep things cheap
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understanding Ok, any idea of how to maximize cutoff point ?
how to analyze quicksort ??
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Analysisso how to analyze quick sort
think how we did mergesort analysis
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Quick sorti = size of left partitionC1 = time to choose pivotC2 = partitioning the set
what do you get?
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what would be the worst case runtime?
can you solve this with the methods from Monday?
( ) ( ) ( )121 −−+++= iNTiTNCCNT
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Telescope
( ) ( ) ( )( ) ( )( ) ( ) ( )
add
ncNTNTcNNTNT
NTTcNNTi
......221
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0
−+−=−+−=
−++==
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what is the big O runtime ??
this was pretty clear before the analysis ☺
its slow-sort ☺
what is the best case
( ) ( ) ( )⎟⎠⎞
⎜⎝⎛ −
+=2
11 NNcTNT
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i = N/2T(N) = 2T(N/2)+cN
For average case if you were to analyze every possible input, when left small, right large
( ) ( )∑−
=
=1
0
1 N
jjT
NiT
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Telescope This!
( ) ( )⎥⎦
⎤⎢⎣
⎡+= ∑
−
=
1
0
12N
jjT
NcNNT
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telescope
( ) ( )
( ) ( ) ( ) ( )
...)1()1(2)1()1()(*
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2*
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0
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−−+−=−−−
−=−−
+=
−
∑
∑
−
=
−
=
NccNNTNTNNTNsubtract
NcjTNTN
telescope
cNjTNTN
Nmultiple
N
j
N
j
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Bottom line
NNNT
NON
NTsimplifies
iCT
NNT N
i
log)(
)(log1)(
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)1(1)( !
0
=
=+
+=+ ∑
+
=
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Wrapping up sortingconsider three elements
a,b,c
If we want to sort these three, what possible ordering can there be ?
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comparisonsif we compare a & b
left if a<bright if b<a
what are subtrees ?
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Decision Treedecision trees portrays the comparisons made by some algorithm
can imagine a tree for quick sort
we can use it represent any comparison based sorting algorithm
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so lets discuss some tree ideas, and apply them back to sorting routines
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Lemma 1A binary tree of depth d has at most 2d
leaves
can you prove this ??
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Lemma 2A binary tree with L leaves must have a depth of at least log(L)
proof should be obvious
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Lemma 3Any sorting algorithm that uses only comparisons requires at least log(N!) comparisons in the worst case
since N! ordering possibilities
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Theoremany sorting algorithm that only uses comparisons requires Θ(N log N) comparisons
can you show this ?
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log(N!) = log (N * N-1 * N-2 * N-3 …log of product = log of sums
log(N!) = log(N) + log(N-1) + log(N-2)..(drop n/2 terms) ≥ log(N) + … + log(N/2)≥ n/2 log(N/2) = ΘN log N
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so how to get past the N log N barrier ??
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more informationif you have extra information you can break the n log n barrierknowing the range, we can use other sorts
bucket sort
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External Sortssay you can only handle 100 items at a timeneed to sort 1000
general case: many times the number of instances to sort will not fit into memory
Strategy:any ideas ?
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switch gearslets switch gears for a second
Question:I want to be able to give you a bunch of items and you should say which set the items belong to (given this information)
should be able to add items to setsshould be able to lookup an item’s set
I want to be able to do it quickly
Example: search results: want to be able to tell you what broad categories a search results can be divided by
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any implementation ideas ?
runtime ?
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equivalenceA relationship R is defined on a set, if for every pair (a,b) we can answer aRb as true of false
Equivalence relationship:1. reflexive
1. aRa for any A
2. symetrics1. if aRb then bRa
3. transitive1. if aRb and bRc then … aRc
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one implementation idea:
use large matrix and mark if relationship exist
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lets do a quick example of a bunch of join’s using a matrix
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Equivalence classan equivalence class of an element a in S is the subset of S that contains all elements related to a
what are the equivalence classes in the previous example?
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online vs offlinesome DS operations can be online some offline
Offline:get all information, and then can process
Online:need to deal with the information before continuing
Example:paper exam vs oral exam
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Equivalence Class ADTUnion(a,b)
merges 2 equivalence classes
Find(a)retrieves equivalence class containing a
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Analyzing Equivalence classeswhen doing the analysis here we will be interested in series of M operations
lets go to implementationany ideas ?
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Arraysuse an array and store name of class at each position
example of union
what is the running timefind?union?
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running timesso find we can do in O(1)
merge, worst could be O(N)for M merges on N items
O(NM)M can’t exceed Nso O(N2)
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linked listseach member will be in its own list
merge ?find ?
will come back to this next week
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Review timeLets start a general review
make sure you are familiar with everything covered so far
address some of the questions seen
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runtimesRuntimes are a rough way of judging algorithms
what is Big-O
why classes of functions ?
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Recursionunderstanding recursion
tail recursion
when to use
when not to use
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what is a DS ?
what are ADT ?
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LISTwhat is the list ADT ?
operations ?runtimes ?
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Listsarrays
linked lists
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queueswhat is a the Queue ADT ?
operations ?runtimes ?
Priority queue ADT ?operations ?runtimes ?
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how do the number of links affect the linked list class ?
single vs double linked
what if we added a mid point link
what is we added a bunch of others
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Treeswhat is the Tree ADT ?
operationsruntimes ?
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Treescomplete treesbinary treesBSTBalanced BST
AVLRed-black
lazy deletiontree traversal algorithmsexpression treesB+ treeshuffman trees
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heapswhat are heaps ?
what DS being implemented ?
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hashtablewhat is the hash table ADT ?
operations ?runtimes ?
what issues need to be dealt with during operations ?
why not issue with Lists ?Extendible hashing
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sortingbasic sorts
bubbleinsertselectionrandom
better onesheap sort
Better yetmergesortquicksort
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Sample Exam
lets do some sample questions together
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Sample 1An algorithm takes 0.5 ms for input of size 100. How long will it take for input of size 500 if the runtime is of the following (assume low order terms are negligible). Show work.
a) Linearb) O (N log N)c) Quadraticd) Cubic
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Sample 2Programs A and B are analyzed and found to have worst-case running times no greater than 150N log N and N2, respectively. Answer the following if possible:
a) Which program has the better guarantee on the running time for large values of N (N > 10,000)?b) Which program has the better guarantee on the running time for small values of N (N <100)
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Sample 3Suppose we want to add an operation FINDKth to our toolbox of binary tree operations. This operation returns the kth smallest item in the tree. Assume all items are distinct (no repeats) Explain how you would modify the binary tree structure to support FINDKth. This operation also must run in O (log N) average time , without sacrificing the time bound an any other operation currently in the tree.
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Sample 4Show the result of inserting the keys 1011-1101, 0000-0010 , 1001-1011, 1011-1110, 0111-1111, 0101-0001, 1001-0110, 0000-1011 , 1100-1111, 1001-1110, 1101-1011 , 0010-1011, 0110-0001, 1111-0000, 01110-1111 into an initially empty B-tree structure with M=3, L=3. These are 8 character keys , the dash makes them easier to read.
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any other questions ??
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Nextdo all the readingthe exam is open brain/notes/book closed general internetwill post when test is ready
please email/aim but will only answer up to test time
Good luck